I have the following questions. Let $$ 0\to A\to G\to Q\to 1 $$ be a central group extension. Also let $id$ be the cohomology class in $H^1(A,A)$ corresponding to the identity homomorphism ($A$ is always treated below as a trivial module).
Now consider the cohomological Lyndon-Hochschild-Serre spectral sequence associated to this extension with coefficients in $A$. Then:
Assuming that $A=\mathbb{Z}/p$, is it true that $d_2(id)\in H^2(Q,A)$ is equal to $\beta$, the cohomology class of this extension (as defined in Weibel's Introduction to Homological Algebra, section 6.6)?
If the first claim is true, would it still be true for all abelian groups $A$? Or at least finitely generated ones?
Thanks!
EDIT: As per request, I am adding some context. I know that the needed differential is the transgression. Now if I think about LHS spectral sequence as the spectral sequence associated to the fibration $$ NA\to NG\to NQ, $$ then I have the diagram (see McCleary's Introduction to Spectral Sequences, Section 6.2): $\require{AMScd}$ \begin{CD} 0@>>>H^2(NQ,\ast;A)@>>>H^1(NQ,A)@>>>0\\ @VVV @V p_0^\ast VV @VpVV @.\\ H^1(NA,A)@>d>> H^2(NG,NA;A)@>>> H^2(NG,A). \end{CD} Here rows are exact and the maps $p_0^\ast$ and $p^\ast$ are induced from the map of pairs $(NG, NA)\to (NQ,\ast)$. Now in order to know that $d_2(id)=\beta$, using the transgression I need to know that $p_0^\ast(\beta)=d(id)$. And this is what I cannot see, given the usual definition of $\beta$: I am choosing set theoretic section $\eta\colon Q\to G$ such that $\eta(1)=1$, and I define $$ \beta(g,h)=\eta(g)\eta(h)\eta(gh)^{-1}. $$ If someone knows a reference, a reference would do for me. Thanks!